Berg Em Anna Dv in Phys 2003

8/13/2019 Berg Em Anna Dv in Phys 2003

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Quasi-two-dimensional Fermi liquid properties of the unconventional

superconductor Sr2

RuO4

C. BERGEMANN1*, A. P. MACKENZIE2, S. R. JULIAN3,

D. FORSYTHE3 and E. OHMICHI4

1Cavendish Laboratory, University of Cambridge, Madingley Road,

Cambridge CB3 0HE, UK and Laboratorium fu r Festko rperphysik,

ETH Ho nggerberg, 8093 Zu rich, Switzerland2School of Physics and Astronomy, University of St. Andrews,

North Haugh, St. Andrews, Fife KY16 9SS, UK3Cavendish Laboratory, University of Cambridge,

Madingley Road, Cambridge CB3 0HE, UK4Department of Physics, Graduate School of Science, Kyoto University,

Kyoto 606-8502, Japan and Institute for Solid State Physics,

University of Tokyo, Kashiwa 277-8581, Japan

[Received 29 April 2002; revised 18 August 2003; accepted 3 September 2003]

Abstract

In this paper, we review a large set of experimental data acquired over the pastdecade by several groups, and demonstrate how it can be used to construct adetailed picture of the low-temperature metallic state of the unconventionalsuperconductor Sr2RuO4. We show how the normal state properties can beconsistently and quantitatively explained in terms of Landau quasi-particlesmoving on a quasi-two-dimensional Fermi surface. Besides presenting our fulland extensive data sets, we explain the details of some novel data analysis toolsthat can be used within the general context of quasi-two-dimensional metals. Wethen use the experimental Fermi surface and band dispersion to reassess severalissues relevant to the unconventional superconductivity in Sr2RuO4, such as thespin-fluctuation spectrum, quasi-particle renormalization, interlayer dispersionand pressure dependence.

Contents page

1. Introduction 6411.1. Unconventional superconductivity in Sr2RuO4 6411.2. Studies of the normal state 6421.3. Doping and pressure studies 6431.4. Scope of this review 643

2. A first view of the electronic structure of Sr2RuO4 6442.1. Basic considerations 644

2.1.1. Electronic configuration of Ru4 6442.1.2. Band formation 644

2.2. Parameterization: tight-binding versus geometric 6473. Magnetic oscillations in quasi-2D metals 650

*Author for correspondence. e-mail: [email protected]

Advances in Physics, 2003, Vol. 52, No.7, 639725

Advances in Physics ISSN 00018732 print/ISSN 14606976 online # 2003 Taylor & Francis Ltdhttp://www.tandf.co.uk/journals

DOI: 10.1080/00018730310001621737


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3.1. de Haasvan Alphen oscillations 6513.1.1. Electron movement in a magnetic field 6513.1.2. 2D electron gas at constant chemical potential 6523.1.3. Real metals 6543.1.4. Quasi-2D metals 654

3.1.5. Beats in the simple Yamaji scenario 6563.1.6. Beats in a more general scenario 6563.1.7. The influence of spin Inormal spin-splitting 6583.1.8. The influence of spin IIanomalous spin-splitting 6593.1.9. Finite temperature and impurities 659

3.2. Angle-dependent magnetoresistance oscillations 6603.2.1. Introduction 6603.2.2. Semiclassical DC magnetoconductivity 6613.2.3. Quasi-periodicity 6613.2.4. Analytical results for a circular Fermi contour 662

3.3. Cyclotron resonance 6643.4. Quasi-particle masses 665

4. Experimental techniques and analysis procedures 6664.1. dHvA techniques 666

4.1.1. Field modulation and cantilever deflection 6664.1.2. Sample sweep and beat pattern 668

4.2. Envelope extraction 6694.3. Numerical approach to AMRO, cyclotron resonance, and

dHvA in quasi-2D systems 670

5. Experimental results and analysis 6715.1. Basic properties 6715.2. Basic Fermi surface topology 6725.3. In-plane Fermi contour 673

5.3.1. From ARPES 6735.3.2. From AMRO 6745.4. Full warping analysis 675

5.4.1. de Haasvan Alphen data 6755.4.2. sheet 6805.4.3. b sheet 6825.4.4. sheet 686

5.5. Mean free path 6875.6. Visualization 6885.7. Consistency checks 688

5.7.1. Resistivity anisotropy 6885.7.2. AMRO 689

5.8. Enhancement of the thermodynamic cyclotron mass 6895.9. Effect of hydrostatic pressure 691

5.9.1. Pressure dependence ofTc 6915.9.2. Fermi liquid behaviour 6925.9.3. Fermi surface geometry 6925.9.4. Quasi-particle renormalization 6935.9.5. Interlayer dispersion 693

6. Discussion and conclusions 6956.1. Basic aspects of the interlayer dispersion 695

6.1.1. Significance and interpretation of anomalousspin-splitting 698

6.1.2. Tight-binding fit to the dHvA Fermi surface 700

6.2. Fermi surface nesting andq-dependent susceptibility 7006.3. Vicinity of the van Hove singularity 7066.4. Effect of charge doping by chemical substitution 708

6.4.1. Ca2ySryRuO4 7086.4.2. Sr2Ru1yTiyO4 7086.4.3. Sr2yLayRuO4 709

C. Bergemann et al.640


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6.5. c-Axis dispersion and superconductivity 7096.6. Comparison with electronic structure calculations 7126.7. Future work 714

Acknowledgements 715

Appendix A. Interpretation of cyclotron resonance experiments 715

Appendix B. Origin of the mass enhancement 719

References 721

1. Introduction

The layered perovskite Sr2RuO4 first received widespread attention due to the

discovery of superconductivity by Maeno and coworkers in 1994 [1]. Since then,

a decade of intensive research has established the unconventional nature ofthat superconductivity, although a host of open questions remain concerning its

microscopic origin [2]. In parallel with work on the superconducting state, consider-

able experimental effort has been expended on understanding the metallic state

from which the superconductivity condenses. Here, progress has been more rapid,

to the extent that Sr2RuO4 has been established to be a quasi-two-dimensional

(quasi-2D) Fermi liquid, whose quasi-particle properties have been measured with

high accuracy [3, 4].

In this paper, we aim to describe the low-temperature normal state properties

of Sr2RuO4 in detail, and to demonstrate how this information was obtained

from experiments. The work is based on an extensive set of experimentaldata (mainly the de Haasvan Alphen (dHvA) effect and magnetotransport) that

relied on the existence of extremely high-purity samples, with mean free paths

approaching 1 mm.

In the course of analysing the data, it was necessary to develop analytical tools

which will be more widely applicable than just to Sr2RuO4. The dual aims of this

article are therefore to give as comprehensive a picture of the Fermi liquid properties

of Sr2RuO4 as we can, and also to present what we hope will be a physically

transparent description of the analysis methods that enabled the construction of that

picture. We then examine the extent to which the extracted information allows us toobtain insight into the relationship between the normal and superconducting states

in this fascinating material.

1.1. Unconventional superconductivity in Sr2RuO4To motivate the study of the metallic state of Sr2RuO4 we start with a very brief

and non-exhaustive outline of its unconventional superconducting properties. We

will not be able to scratch much below the surface, however, and for a more in-depth

discussion, the reader is referred to the recent review article on this topic [2] and the

original research articles referenced therein.Sr2RuO4 has a critical temperature of Tc 1:5 K (in the clean limit) and

originally attracted interest because it possesses a layered perovskite structure like

the high-Tc cuprates and is therefore one of the very few perovskite oxide super-

conductors containing no copper. Further studies then indicated that the properties

Quasi-2D Fermi liquid properties of Sr2RuO4 641


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of Sr2RuO4 itself deviate strongly from the BCS standard model of s-wave

superconductivity; there is now good evidence that Sr2RuO4:

1. shows spintriplet pairing,

2. exhibits time-reversal symmetry-breaking of the orbital part of the pair

wavefunction,

3. possesses line nodes in the gap function,

although the latter two experimental facts are in some ways difficult to reconcilewe

will return to this issue later.

Concerning the superconducting mechanism in Sr2RuO4, the prevailing

opinionalthough by far not the only oneis that it is mediated by spin

fluctuations. However, there appear to be two different schools of thought as to

whether the most significant pairing channel arises from low-q or high-q contribu-

tions to the spin fluctuation spectrum, and this question is inextricably linked with

the details of the underlying electronic structure in the normal state. Again, we willdeal with these issues at a later stage, after establishing the Fermi liquid properties

of Sr2RuO4.

1.2. Studies of the normal state

While the verdict is still open even on the qualitative aspects of the super-

conductivity of Sr2RuO4, we are in the fortunate position that its Fermi surface

and quasi-particle spectrum can be studied experimentally in considerable detail.

Information about the normal state has been contributed by a succession ofexperimental and theoretical papers, starting from early local density approximation

(LDA) band structure calculations [58], dHvA data ranging from early confirma-

tions of the basic Fermi surface topography [3] to a detailed survey of the interlayer

dispersion [4], Hall effect [9, 10] and magnetoresistance data [11], photoemission [12],

and microwave absorption measurements ([13] and others).

We are now in a position to compile this knowledge, which was up to now

scattered across a large number of original papers, and to arrive at a near-complete

quantitative description of the low-temperature normal state of Sr2RuO4. Such solid

quantitative input cannot come from theory alone: while LDA calculations [58] can

serve as a guideline, their numerical predictions can deviate from the experimentalvalues of, for example, the Fermi surface cross-sectional areas by 10% or more (see

table 1 in section 2.2 and the discussion in section 6.6). Such deviations are critical if,

as is the case for Sr2RuO4, the normal state is in the neighbourhood of several

different magnetic instabilities. The potential effects of these deviations are particu-

larly amplified on the sheet, due to the vicinity of the van Hove singularity.

We hope that the results that we have compiled will serve as quantitative input

for theoretical attempts to comprehend the superconductivity. Sr2RuO4 takes on a

special position in correlated electron physicsrivalled perhaps only by the heavy

fermion superconductor UPt3 (see [14] for a recent review)in that it is an

established unconventional superconductor, with a conventional low-temperaturenormal state. Here we take the term conventional to describe a metal that conforms

to Fermi liquid theory, the standard model of the metallic state. This is strikingly

different from the situation in the high-Tc superconducting cuprates, say, where the

normal state appears to pose an equally formidable challenge in that pseudo-gaps,



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stripes, Fermi surface hot spots, non-Fermi-liquid resistivity power laws, etc. have

opened up a Pandoras box of competing theories (for a recent review, see e.g. [15]).

1.3. Doping and pressure studies

To extract further information on the relation between the normal stateand superconductivity, one needs to examine Sr2RuO4 not just under ambient

conditions, but also in the neighbouring phase space. A methodical investigation

of the phase space around a stoichiometric compound can usually proceed either by

chemical doping or by application of pressure.

It has been reported how hydrostatic pressure affects both the superconducting

properties and the Fermi surface in Sr2RuO4 [16], and this should provide a

definitive testing ground for theoretical models of superconductivity in Sr2RuO4.

Several doping studies have also been carried out, mainly by the Kyoto group, with

Ca and La substituting Sr [17, 18], and Ti and Ir substituting Ru [19, 20]. Owing to

the rapid suppression of non-s-wave superconductivity in Sr2RuO4in the presence of

(even non-magnetic) impurities [21], these results relate more to the normal state

physics of Sr2RuO4. In this article we will provide additional quantitative analysis,

both for the pressure and doping studies.

1.4. Scope of this review

We begin the review with an outline of the qualitative aspects of the electronic

structure of Sr2RuO4. Like many transition metal oxides, it can be understood in

terms of the tight-binding approximation applied in this case to the d electrons on

the Ru4 sites. We then present an introduction to the most relevant experimental

techniques for the study of quasi-2D metals: quantum oscillations, angle-dependent

magnetoresistance oscillations, and microwave resonance. Particular emphasis is

placed on a new method of using dHvA amplitude information for high-precision

studies of the interlayer dispersion.

Next we present the available data, with a focus on dHvA experiments since

this is both the main area of the authors expertise, and also, we believe, the technique

that has brought about the most significant quantitative progress. We deduce a near-

complete picture of the normal state properties, in particular the Fermi surface and

the quasi-particle dispersion near the Fermi energy, and focus on a few key issues of

possible relevance in the context of superconductivity:

the nesting properties of the Fermi surface, and the q-dependent spinfluctuation spectrum;

the quasi-particle renormalization; the interlayer dispersionat first sight, this might seem surprising as Sr2RuO4

has traditionally been thought of as an essentially two-dimensional problem,

but the horizontal ring nodes in the gap function that have been proposed

theoretically [22, 23] and experimentally [24, 25] would immediately signify the

relevance of interlayer effects for the superconductivity;

the correlation between the pressure dependence of both normal state andsuperconducting properties.

Finally, we compare the experimental results with band structure calculations,

and touch upon the evolution of the normal state of Sr2RuO4on charge doping (by

chemical substitution).



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The scope of this article is quite broad, and not every reader will be interested in

its entirety. For guidance,

readers chiefly interested in the physics of Sr2RuO4 should focus on section 6and appendix B, and also browse through sections 2 and 5;

the experimental data relevant to the normal state properties of Sr2RuO4 arepresented and analysed in section 5 and appendix A;

newcomers to the field of quantum oscillations seeking first guidance arereferred to the first parts of section 4 for an introduction to the physics, and

section 4 for a primer on the experimental and analysis techniques; and

researchers who want to learn in detail about the way we extract electronicstructure information should read thoroughly through the later parts of both

sections 3 and 4, and work through section 5 for an in-depth case study.

2. A first view of the electronic structure of Sr2RuO42.1. Basic considerations

2.1.1. Electronic configuration of Ru4+

The physics of transition metal oxides is, in a first approximation, mainly the

physics of the d-electron shell [26]: if we take strontium and oxygen to have the usual

nominal valencies Sr2 and O2, the ruthenium ion Ru4 in Sr2RuO4is left in a 4d4

electronic configuration. In the octahedral crystal field environment of the layered

perovskite structure of Sr2RuO4, the energetically low-lying d levels are the three

orbitalsnominally referred to as dxy, dxz, and dyzthat belong to the t2grepresentation of the octahedral group (see figure 1). These orbits can accommodate

six ("#) electrons and are partially filled in the 4d4 configuration in Sr2RuO4.

2.1.2. Band formation

In general, metallic behaviour is fairly rare among the transition metal oxides,

and more often than not the d electrons remain localized, frequently forming a

magnetic insulating state [26]. This is mainly due to the large on-site (Hubbard)

repulsion and the small interatomic overlap, both arising from the small orbital

radius of the d electrons, particularly in the 3d series.

There are exceptions, though, and Sr2RuO4 is one of several ruthenium oxide

systems where the d electrons are itinerant. From the simplest point of view, the

Fermi surface is then formed in a tight-binding fashion from the dxy, dxz, and dyzatomic orbitals on the ruthenium sites. There is also some antibonding admixture of

the oxygen p orbitals (see for example figure 2) which, unlike the situation in most

other oxide metals, has significant weight at the Fermi energy [6]. The layered

structure of Sr2RuO4 prevents strong overlap of the orbitals along the c-axis, hence

the electronic structure is, to a first approximation, two dimensional.

Both the dxz and dyz atomic orbitals are odd under the mirror operation z! z,while dxy is even; therefore we expect the bands formed by the dxy and dxz,yzorbitals, respectively, not to mix, at least not at kz0. Considering the quasi-two-dimensional nature of Sr2RuO4, this assumption can be extended to the whole of

the Brillouin zone: the dxy and dxz,yz bands form subsystems that are very weakly

coupled.

A simple tight-binding model now demonstrates how the Fermi surface of

Sr2RuO4 can be constructed qualitatively: as the spatial extent of the dxz atomic

orbitals is mostly in thexz-plane, we expect sizeable orbital overlap mainly between



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nearest neighbours along thex-axis. Correspondingly, the band energy only depends

on kx, and the xz part of the Fermi surface consists of an open sheet more or less

perpendicular to the x-axis. Similarly, the yz part is an open Fermi surface sheet

perpendicular to the y-axis. The xz and yz states hybridize for kx, ky6 0, and theFermi surface sheets regroup to form the structure shown in figure 3: an electron

cylinder (called b) centred around the line kxky 0, and a hole cylinder(called) centred around the corners kxky p=a of the Brillouin zone.

a

c

Ru4+

e

t

Oxygen

Ruthenium

Strontium

g

2g

4d

Figure 1. In the octahedral environment of the Ru4 ions in the layered perovskite crystallattice of Sr2RuO4 (left), the 4d levels experience crystal field splitting into low-lying t2g(dxy, dxz, dyz) and higher eg (dx2y2 , dr23z2 ) states (right). The low-temperature crystallattice parameters [27] area 3:86 A andc 12:72 A .

a

Ru Ru

RuRu

O

O

O

O

++ +

+

+

+

+

+

+

+

+ +

Figure 2. The completely antibonding state atkx, ky p=a, p=a at the top of the dxyband (looking down the c-axis).



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The dxy orbitals, on the other hand, are able to hybridize in both in-plane

directions, and one can even expect sizeable next-nearest neighbour overlap along

the orbital lobes, i.e. diagonally. Therefore, the xy part of the Fermi surface forms

some sort of cylinder, closed in the kxky-plane but open along kz. Without further

study, it is difficult to assess whether this cylinder should be an electron orbit centred

around the line or a hole orbit centred around the Brillouin zone corners.

Band structure calculations [58] settled this question in favour of the electron

cylinder (called ) and generally agreed with the qualitative picture laid out above.

They also provided some extra information, notably the bare bandwidth. Thedetailed results are still model-dependent, though, and the effects of strong electronic

correlations are not easily included.

The early dHvA work [3] confirmed the predictions from band structure

calculations. The quantitative analysis suggested that the conduction electrons in

Sr2RuO4 are very evenly distributed amongst the three orbits, with every orbital

flavour carrying almost exactly 4=3 electrons. The cylinder appears to pass right

through the hybridization gap between the and b cylinders, avoiding the issue of

the extent to which the dxy and dxz,yz bands mix near the Fermi energy (for further

discussion, see section 5.4.3).

The simple, two-dimensional Fermi surface shown in figure 3 summarizes allthese findings and completes our poor mans survey of the electronic structure of

Sr2RuO4. While this undeniably serves as a good starting point, it appears today

that much more detailed knowledge is needed to account quantitatively for the

unconventional superconductivity in Sr2RuO4. This is particularly true because the

peculiarities of the Fermi surface give rise to a number of competing instabilities

which are thought to play a key role in the superconducting pairing mechanism

(see section 6.2).

We will therefore spend the remainder of this paper trying to improve on figure 3,

focusing on a refinement of the in-plane Fermi contour, details of the interlayer

dispersion, and the quasi-particle mass enhancement and its origin. Besides the

obvious quest for the mechanism of superconductivity, one good reason for doing

all this is to demonstrate the level of precision with which it is now possible to

understand a correlated electron metal. The crystal quality of Sr2RuO4, with mean

free paths approaching 1 mm, enables us to map the Fermi surface and electronic

X

M

dxz

dyz

Figure 3. Qualitative sketch of the Fermi surface in Sr2RuO4. The right panel shows how

the dxz and dyz bands regroup to form the and b surfaces.



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structure of this compound to a level of detail rarely seen outside the realms of the

pure elements.

2.2. Parameterization: tight-binding versus geometric

Before we attempt to quantify the electronic structure of Sr2RuO4, it is helpfulto discuss how a convenient parameterization of this information can be achieved.

From a theoretical point of view, perhaps the most useful approach is the tight-

binding formulation: the deviations from the exact atomic potential lead to an

overlaptijRbetween an atomic d orbital at the origin (where the subscripts iand jdenote the orbital flavour: xy, xz, or yz) and another one at the site R. The energy

bands k are obtained as the eigenvalues of the Hamiltonian matrix

Hij

XRtijR cosk R 1

where the sum runs over all lattice vectors R. The Fermi surface is then extracted

as the equal energy contour at the right filling level of four electrons per Brillouin

zone. Indeed, LDA band structure calculations are, in practice, often approximated

by tight-binding expressions, and within this model, the overlap matrix elements

tijRyield a complete description of the electronic structure. We therefore might betempted to try a fit of the tijR from our dHvA data.

Table 1 lists the parameters tijR most commonly used for Sr2RuO4 in theliterature [2830], as inferred from LDA band structure calculations [7], not

experiment. Two obvious deficiencies are that the xz=yz hybridization t23a, a, 0 isnot given correctly, and that the Fermi surface cross-section for the sheet is off byat least 10% compared to the dHvA value. We will further compare experiment and

LDA calculations in section 6.6.

While the latter point demonstrates how dHvA can help to adjust the parameters

in a tight-binding model, there are, however, two reasons why one has to be cautious

when directly inferring the tight-binding overlap integrals from quantum oscillation

experiments:

1. Thebareelectronic bandwidth is inaccessible by dHvA experimentsonly the

states near the Fermi surface are probed, i.e. one can infer the Fermi surface

geometry and the renormalizedbandwidth.

2. In a strongly interacting system, energy bands and thetijR lose their meaningaway from the Fermi surface.

When interpreting experiments, we will therefore initially parameterize the

electronic structure information obtained from our data in a fashion relating more

directly to the Fermi surface geometry itself, in that we expand the local Fermi

wavevector of the three Fermi surface sheets in cylindrical harmonics [4]:

kF, z X

, 0k

csf g csf g

cos

sin

n oz cos

sin

n o 2

where z ckz=2 runs fromp to p over the height hBZ of the Brillouin zone (c isthe height of the body-centred tetragonal unit cell), and is the azimuthal angle ofk

in the kxky-plane.

We can leave out the sin z terms because they do not conform to the kz! kzmirror symmetry of the Brillouin zone of Sr2RuO4. Additional symmetries further



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Table 1. Tight-binding parameters used in the literature for Sr2RuO4. The subscripts refer to the ruthefifth column gives the calculated values for the dHvA frequency of the orbit and the experimental in each case of 10% or more. The second-to-last row shows the best fit to the experimental Fermi su

Ref. t110a t11a; 0; 0 t11a; a; 0 F t220a t22a; 0; 0b t220; a; 0 [28] 0.4 eV 0.4 eV 0.12 eV 16.1 kT 0.3 eV 0.25 eV [29] 0.5 eV 0.44 eV 0.14 eV 16.9 kT 0.24 eV 0.31 eV 0.045 eV [30] 0.5 eV 0.44 eV 0.14 eV 16.9 kT 0.24 eV 0.31 eV 0.045 eV

Section 6.1.2 0.62 eV 0.42 eV 0.17 eV 18.23 kT 0.32 eV 0.30 eV 0.026 eV [3]c DHvA experiment: 18.66kTc

aMeasured from F.bIntroduced unphysically as constant hybridization t230; 0; 0:cNote the small experimental deviation from integer filling (cf. section 5.2) which we have corrected prior to fitting (see


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restrict the number of possible non-zero k. For the b and cylinders, centred on

the line, the fourfold rotation symmetry of the Brillouin zone, along with the

! mirror symmetry, shortens the general expansion in equation (2) to

kF, z X, 0

mod40

kcos zcos ( and cylinders): 3

Similarly, thecylinder is centred on the X line which runs along the corners of the

Brillouin zone. The twofold rotation symmetry imposes even , and the ! p mirror symmetry along with the z! z p, ! p=2 screw axis (see figure 4)further restrict the terms in the expansion: for mod 4 0, i.e. for contributionswith fourfold symmetry, only double warping with even is allowed. In addition,

contributions with mod 4 2 that break the fourfold symmetry are permitted butmust have odd and carry the factor sin (rather than cos ). Formally, we can

then write equation (2) as in

kF, z X

, 0 mod40

even

kcos zcos X

, 0 mod42

odd

kcos zsin ( cylinder):

4In equations (3) and (4), we have left out the s=c superscript to the k because the

symmetry requirements have made it unambiguous whether they refer to a sin or

cos term. The in-plane Fermi contour is given by the parameters k0, while theinterlayer dispersion corresponds to the warping parameters k with >0, which

-LineX-Line

c4/

a2/

Figure 4. Due to the peculiar body-centered tetragonal Brillouin zone stacking, Fermicylinders in Sr2RuO4 possess two distinct symmetries, depending on whether they arecentered on the or X line. The Brillouin zone exhibits a simple fourfold symmetry (withmirror planes) around the line; around the X line, on the other hand, the two-foldrotation symmetry is augmented by the screw axisz!z p; ! p=2.



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are taken to be much smaller than the average Fermi wavevector k00 in quasi-2D

metals like Sr2RuO4. The different warping contributions are illustrated, and the

symmetry requirements in Sr2RuO4 are summarized, in table 2.

The tight-binding parameterization tijR of the electronic band structure andthe geometric description k of the Fermi surface are, of course, intimately related.

One can calculate (numerically, usually) the Fermi surface geometry directly fromthe equal energy contours of the tight-binding band structure. Conversely, one can

try and solve the inverse problem, inferring tight-binding parameters from details of

the Fermi surface, keeping in mind the caveats listed on page 647. In particular, the

tight-binding parameters have to be normalized against the overall bandwidth.

We will pursue this Fermi surface fitting with suitable tight-binding parameters in

section 6.2.

3. Magnetic oscillations in quasi-2D metals

In this section, we give an outline of the quantitative description of thephenomenade Haasvan Alphen (dHvA) effect, angle-dependent magnetoresis-

tance oscillations (AMRO) and microwave absorption effects, in particular periodic

orbit resonancethat have contributed the most significant information about the

details of the Fermi surface and band dispersion of Sr2RuO4.

Of these, AMROs and microwave conductivity are very much related, since

they can both be understood semiclassically, arising from the cyclotron motion in

momentum space. Both dHvA and microwave measurements also give information

about the quasi-particle masses in the material, and we present a short outline of what

exactly is meant by the quasi-particle mass in which context, and how these masses

relate.Most of what is presented in this section is not, in itself, new, but it does go

beyond the content of most contemporary textbooks. A comprehensive and very

detailed treatment of the dHvA effect has of course been given by Shoenberg [31]a

book which is now, unfortunately, out of print. Other in-depth treatments of the

Table 2. Illustration of the warping contributions k for the different values of and (added on top of a large average Fermi wavevector k00). The last two rows showwhether these warping parameters are allowed () or forbidden () by the Brillouinzone symmetry for the sheet and the = sheet, respectively. For the k21 symmetrysee also figure 33 later in this article. (Artwork reprinted from [4] with permission

from the American Physical Society 2000.)

k40 k01 k02 k21 k41 k42cos4 cos z cos2z cos zsin 2 cos zcos 4 cos2zcos 4

xkk

z

;



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dHvA effect have appeared in the Shoenberg Festschrift[32]. AMROs and periodic

orbit resonances in quasi-2D metals are a more recent development, and their

treatment is scattered across a number of original papers [3335] and monographs

[36]. It certainly appears useful to give a short yet comprehensive introduction to

these effects in this review, since their detailed understanding is absolutely needed forthe interpretation of the experimental data sets.

Our experimental view of the Fermi surface of Sr2RuO4also relies on the results

of a few other experiments, notably angle-resolved photoemission spectroscopy

(ARPES) and aspects of the nuclear magnetic resonance (NMR) technique.

However, here the situation is reversed in that contemporary textbooks (see, for

example, [37] and [38]) describe these experiments in detail, while on the other hand

they did not take the centre stage in the determination of the electronic structure of

Sr2RuO4. For that reason, the physical basis of these techniques will not be discussed

in detail in this article, although we will make use of some of the results that they

have yielded on Sr2RuO4.

3.1. de Haasvan Alphen oscillations

3.1.1. Electron movement in a magnetic field

Classically, a freely moving electron in a homogeneous magnetic field Bunder-

goes a helical motion along the field lines in real space (figure 5(a)). Viewed in

momentum space, the electron traverses a circular orbit (figure 5(b)). The Lorentz

force relates the movement in real space and momentum space by

dp e dr ^ B 5

i.e. the momentum space movement is at 90 to the real space one and scaled up by afactor eB.1

In quantum mechanics, the solutions n of the Schrodinger equation (figure 5(c))

in the presence of a magnetic field (see, e.g. [39]) reflect this cyclotron motion, with a

radial part peaked at a distance r2n 2n 1hh=eB from the center of the motion.Their energy is En n 12hheB=m; their degeneracy, achieved by spatial translation,is independent ofn but proportional to the sample volume and to B.

The orbital area Arn (in the plane perpendicular to B) is pr

2

n 2n 1p

hh=eB.Using equation (5), the orbital radius kn and area Akn in momentum space become

Akn pk2ne2B2Ar n 2peB

hh 6

with 12

in this case. That is, the cyclotron orbit area in k-space is quantized in

units proportional to B.

Not surprisingly, the basic momentumenergy dispersion is retained, as

En hh2k2n=2m. Here we have neglected the spin (Zeeman) contribution, and we willcontinue to do so until we reach section 3.1.7.

1Here and in the following we only consider the mechanical momentum m_rr and not the canonical

momentum m_rr eA.



14/87

3.1.2. 2D electron gas at constant chemical potential

Let us consider a 2D gas of free, non-interacting electrons. In the absence of

a magnetic field, the electrons take on momentum eigenstates eikr, with energyhh2k2=2m. Boundary conditions restrict the choice ofk, and the Pauli principle limits

each statek to hold no more than two ("#) electrons. In the ground state, i.e. at lowtemperatures, all states with energy below the chemical potential, E< , are filled,

and states withE> are empty. The filled states are bunched inside the Fermi disk

(see figure 6(a)) which has an energy EF and a radius kF around the origin inmomentum space.2

If a magnetic field is switched on, the electrons redistribute into the orbitaleigenstates. Their degeneracy is so high that each orbit of area n 1

22peB=hh in

momentum space holds a macroscopic number of electrons: roughly those that had

k-values lying in between this and the next orbit, of arean 322peB=hh.

So instead of having all the k-states up tojkj kF filled, we now have all theorbitalstates up to knkF occupied (figure 6(b)). As we raise the magnetic field, theradiiknof the orbital states in k-space grow as

ffiffiffiffiB

p , and their energyEnas B, and one

by one they will pass through the Fermi edge at kF andEF (figure 7). As a state does

so, it abruptly becomes vacant (still under the assumption of a constant chemical

potential at T!

0), shedding all electrons in that state, thereby losing a macro-

scopic percentage of electrons in the system.

This macroscopic and discontinuous change in the electron number of the system

occurs every time an orbit crosses the Fermi edge, which is most conveniently

expressed in terms of the area: AF Akn , whereAF is the area of the Fermi disk, orhhAF=eB2pn 12. This demonstrates that the electron number has an oscillatorycomponent sin2pF=B as a function ofinversemagnetic field, with the (de HaasvanAlphen) frequency

FhhAF2pe

: 7

B

px x

py y

pz ||

(a) (b) (c)

Figure 5. Movement of free electrons in a magnetic field: (a) classical, in real space;(b) classical, in momentum space; (c) quantum-mechanical, in real space.

2We take the system to be at constant chemical potential for simplicity, and neglect the small

corrections arising in a constant particle number treatment. The difference, which is visible only under

very special circumstances, is that the chemical potential can oscillate and induce sidebands in the dHvA

frequency spectrum [40, 41].



15/87

As the oscillations are periodic in inverse field, Fhas the units of teslas (or kiloteslas,

kT, not to be confused with the thermal energy kBT).

Due to the discontinuous jump, we can locally approximate the shape of the

oscillations as a saw-tooth, with Fourier components at harmonics Fthat have an

amplitude 1= that of the fundamental.

This oscillation feeds back into most low-temperature thermodynamic and

transport material properties. This is easily seen for the diamagnetic magnetization

induced by the field: semiclassically, the magnetic moment of the orbital state n is

simply its area An, times the cyclotron frequency, times the number of electrons in

the state. That is, the system experiences a discontinuous jump in magnetization

proportional to the jump in electron number, also leading to saw-tooth oscillations

in the magnetization.

(a) (b)

Figure 6. Filled and empty states in momentum space: (a) momentum eigenstates in theabsence of a magnetic field, (b) orbital states in a magnetic field.

(orbits expand with

netic field)

increasing mag-

Numbero

felectrons

B

Figure 7. Discontinuous change of the total electron number of the system as the orbitalstates cross the Fermi edge.



16/87

The oscillation frequency directly measures AF and therefore the Fermi

wavevector and carrier density in this idealized 2D situation. In real metals, the

situation is somewhat more complicated, but the basic concept remains the same.

3.1.3. Real metalsIn a metal, the quantum-mechanical description of the electron movement is of

course much more intricate. We make the standard simplifying assumption that the

metal is a Fermi liquid, that is, in the absence of a field it can still be described in

terms of quasi-particles, with momentum k, charge e, and spin 12, that exhibit an

occupancy discontinuity in 3D k-space: with one or more Fermi surface sheets

separating the filled from the empty states.

In the presence of a magnetic field, the states can again be described as orbital

states in k-space, peaked around an equal-energy contour of the electronic band

dispersion perpendicular to the applied field. The geometrical shape of these orbits

can vary, but their area is again quantized (provided we are dealing with closedorbits). More specifically, equation (7) still holds for large quantum numbers, since it

is a deeper consequence of the BohrSommerfeld quantization rule. Consequently,

our discussion of the depletion of a whole orbit every time it crosses the Fermi level

remains valid also. The major difference to the idealized 2D case is that now the area

AF of the equal-area contour at the Fermi levelthat is, the Fermi surface cross-

sectional area perpendicular to Bis no longer constant. The corresponding local

frequency Fkk hhAF=2pe therefore varies along the momentum component kkparallel to B. Instead of one single oscillation sin2pFkk=B, and its harmonics,these oscillations then have to be integrated over k

k.

For three-dimensional metals, this integration can be simplified in the saddlepoint approximation. The integrand sin2pF=B is rapidly varying, and only theextremal cyclotron orbits lead to a macroscopic magnetization.3 Therefore, the

observed oscillation frequencies give information about the maximum and minimum

cross-sections of the Fermi surface for any given magnetic field direction; in order to

establish the overall shape of the Fermi surface one has to solve the inverse problem

which is usually done with the guidance of band structure calculations. This is the

usual treatment that has been known for decadesfor a review, focusing on the

transition metals, see e.g. [42].

3.1.4. Quasi-2D metals

For a quasi-2D metal such as Sr2RuO4, the Fermi surface consists of weakly

corrugated cylinders. If the difference between their maximum and minimum cross-

sectional areas is not much larger than the area difference 2peB=hh between two

adjacent Landau tubes, the extremal orbit treatment described above becomes

inappropriate. The oscillatory magnetization needs to be calculated as the full

superposition of the individual contributions of the cross-sectional slices of the

Fermi surface.

In the most basic case, a simple corrugation of an otherwise perfectly cylindrical

Fermi sheet leads in this way to a simple beat pattern in the magnetizationobvious

3This rapid variation also averages out the chemical potential oscillations, justifying the use of a

constant chemical potential rather than the more difficult treatment of a constant particle number.



17/87

perhaps, but we will go through the full calculation below. This demonstrates that

small deviations from the perfect cylinder lead to characteristic amplitude modulations

of the dHvA oscillations. Over long (inverse) field spans, they induce a substructure in

the frequency spectrum, but over the shorter field regions accessible in experimental

reality, direct reverse-engineering of the observed amplitude modulations provides

the most powerful probe of the details of the Fermi surface geometry.

The fundamental component of the oscillatory magnetization is given by the fullsuperposition integral

~MM/Z 2p

0

dzsin hhaz

eB

8

whereaz is the cross-section of the orbital area cutting the cylinder axis at z.4If the magnetic field is applied at polar and azimuthal angles 0 and0, this area

can be calculated using

az 1cos 0 I

2p

0

d1

2k2F , z cos 0 tan 0 9

where the angles and are defined as in figure 8 and is the in-plane Fermiwavevector times 2p=hBZ, i.e. times c=2 for the case of Sr2RuO4.

Equations (8) and (9) describe oscillations with carrier frequency

F0

cos 0 hh

4pe cos 0

I 2p0

dk2F 10

with F0 corresponding, of course, to the area of the in-plane Fermi contour without

the warping corrections. The amplitude of these oscillations is modulated by the

interference pattern induced by the corrugation of the Fermi cylinder. We will now

look at this amplitude modulation in some simple cases, to establish some guiding

4The higher harmonics arising from the saw-tooth shape are given by very much the same

expression, except that we use pa, p 2, instead of a.

z

0

0

0

Figure 8. Cross-sectional area of cyclotron orbits in quasi-2D metals.



18/87

principles for the extraction of the Fermi surface parameters in the real life

situation in Sr2RuO4.

3.1.5. Beats in the simple Yamaji scenario

For a simple corrugation of a Fermi cylinder, i.e. kF, z k00 k01cos z inthe cylindrical harmonic expansion, equation (2), the Fermi surface cross-sectional

area integral in equation (9) can be calculated explicitly [34] as in

az pk200

cos 0 2pk00k01J0Ftan 0

cos 0cos z 11

where small corrections of order k201 have been ignored. F is the average in-plane

Fermi wavevectork00 times 2p=hBZ. The dependence of the cross-sectional area a on

the height z in the Brillouin zone remains purely cosinusoidal. At the Yamaji angle

Y

arctan0=F

, this cosinusoidal modulation becomes zero, and all Fermi

surface cross-sectional areas are equal, to first order. Here, 0 2:4048 is the firstzero of the Bessel function J0.

The quantum-oscillatory magnetization of the corrugated Fermi surface shows

beats due to the two differing extremal orbits, as can be calculated explicitly from

equation (8): the oscillatory magnetization is

~MM/ sin 2pF0B

J0

pF0B

; 12

here F0 F0= cos 0 where F0 hhk200=2e is the average on-axis dHvA frequency,and F0 F0J0Ftan 0=cos 0 where F02hhk00k01=e is the frequencycorresponding to the difference between on-axis maximum and minimum areas.

In other words, the dHvA oscillations have the carrier frequency F0 and anamplitude modulation jJ0pF0=Bj. In analogy to the extremal orbit approxima-tion, this can be approximated for small fields byjcospF0=B p=4j, i.e. themodulation manifests itself as a beat pattern with frequency F0.

The beat pattern has a characteristic dependence on the angle0between the field

and the cylinder axis: the beating is most rapid for 0 08(on-axis) and slows downfor increasing angles. At the Yamaji angleY, where all Fermi surface cross-sections

have equal area, the magnetization contributions all interfere constructively to give a

maximum amplitude without beats (cf. figure 9).Conversely, if we observe such a beat pattern experimentally, we can establish

both the magnitude andfrequency of the Fermi surface corrugation. The magnitude

is given directly by the on-axis Fwhich we can obtain from the beat frequency, or

in extreme cases (small F) we can fit it to the beat pattern. The frequency, or the

corrugation repeat unit alongkz, is accessible through the Yamaji angle. In general,

we therefore need full amplitude information versus field and angle (polar and

azimuthal, in the absence of full rotational symmetry in real materials) to extract

the maximum amount of information. The case of Sr2RuO4 provides a fascinating

case study for the application of these ideas, as set out in the experimental section of

this review.

3.1.6. Beats in a more general scenario

For non-trivial warping parameters, the area integral in equation (9) can still be

solved analytically, as long as the in-plane Fermi contour remains circular.



19/87

Generalizing Yamajis earlier treatment [34], a warping component kc

sf g c

sf g then yieldsan area contribution [4]

az 2pk00kcsf g csf g

JFtan 0cos 0

cos

sin

n o0 fz 13

where the function f is either sin or cos, depending on : 1=2fcos

singz for even ,

11=2f sin cosgz for odd .If all significant hopping processes involve nearest-neighbour planes only, i.e. if

only warping parametersk with 1 are present, the overall zdependence of theorbital areas will still look like a purely cosinusoidal modulation. The magnetization

signal then has the same form as in equation (12) but F0 now contains allcontributions from equation (13).

Y

(a)

(b)

(c)

B

B

B

Figure 9. For a simply warped Fermi surface, the two different extremal orbits lead to a beatpattern in the dHvA oscillations. The difference between the maximum and minimumcross-sections gradually changes when the magnetic field is tilted off-axis, until all orbitalcross-sections become roughly equal at the Yamaji angleY[34]. For this field direction,the magnetization contributions from different slices of the Fermi surface all interfereconstructively to give a maximum amplitude without beats. Beyond the Yamaji angle, thebeats appear againuntil the second Yamaji angle is reached (not shown).



20/87

In the case of non-circular in-plane Fermi contours, the integral in equation (9) can

only be solved numerically. Due to the quasi-1D nature of the dxz=yzbands underlying

the and b sheets, this proves to be absolutely necessary for Sr2RuO4. A description of

the algorithm used to perform this computation is given in section 4.3.

3.1.7. The influence of spin Inormal spin-splitting

One also has to consider the effect of the magnetic field: Zeeman splitting of

the spin-up and spin-down electronic energies leads to spin-splitting of the Fermi

surface into two subsheets of slightly different size. The spin flavour parallel to the field

is energetically favoured and so forms the larger of the two subsheets (figure 10(a)).

In a first-order approximation, we can describe the effect using a field-dependent

Fermi wavevector k"#00 k00 00B. Here, 00 represents the rate of the expansion

which is (for a one-band system) proportional to the spin susceptibility.

This expansion feeds back into the Fermi surface cross-sectional areas. In

the simplest case of a cylindrical Fermi surface, the areas change by a"#2pk0000B= cos 0. The dHvA oscillatory signal sinhhA=eB has to be adjustedaccordingly, giving

X

sinhhA 2pk0000B= cos 0

eB sin hhA

eBcos

2pk0000

e cos 0|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}RS

14

Note, therefore, that because they are linear inB, first-order terms in the susceptibility

do not affect the frequency of the observed oscillations, only their amplitude.

The amplitude reduction factor RS is independent of the magnitude (but not the

direction) of the applied magnetic field.

This result and the 1= cos 0 dependence of the argument of the cosine hold quite

generally in quasi-2D metals. The mathematical form of RS implies that the

amplitude actually goes to zero for selected angles 0. In this way, one can infer

the spin susceptibility from dHvA data. There is ambiguity in inverting the cosine,

however, so the data set needs to go to high enough angles 0 to cross at least two

such spin-zeros for a firm determination of the spin susceptibility.

kx,y kx,y

kz kz(a) (b)

B

Figure 10. Illustration of spin-splitting: (a) normal spin-splitting (00) where the magneticfield homogeneously drives apart the spin-up and spin-down surfaces; and (b)anomalous spin-splitting (here: 01) where the field also changes the warping k01.



21/87

3.1.8. The influence of spin IIanomalous spin-splitting

As we will see from the experimental results, it turns out that the ordinary

spin-splitting picture is at odds with the dHvA data for Sr2RuO4. In particular, there

is no B-independent reduction factor RS which goes through zero as the field is

rotated. The experimental situation therefore demands a modification of the picturepresented in the previous section.

We have proposed what is conceptually perhaps the simplest explanation which

also, empirically, describes the data with outstanding precision [4]. In the anomalous

spin-splittingscenario, we take not just the Fermi wavevector k"#00 itself to vary with

B, but also the other warping parameters, as in k"# k B.This would correspond to the underlying electronic band structure being flatter

at some points on the Fermi surface than at others; this situation is depicted in

figure 10(b). In itself, this proposition is certainly not too exotic an approach.

However, as we will see in section 6.6, the effect is unlikely to originate from the

bare (LDA) band structurebut it might enter via the Stoner factor or otherrenormalization effects relevant for the spin susceptibility.

Note that all terms in this anomalous spin-splitting remain linear in B. As such,

this has to be distinguished from the effects of a non-linear susceptibility (higher-

order terms inB becoming important). Such non-linear susceptibilities can introduce

additional, field-dependent corrections to the measured frequencies [43] but appear

to be negligible for the case of Sr2RuO4, cf. the discussion on p. 707.

3.1.9. Finite temperature and impurities

The real power of dHvA is that it gives information not only about the Fermi

surface geometry, but also about the electron masses and the band renormalization

near the Fermi energy. Although well-known to specialists in the technique, this

fact is surprisingly little-known in the broader research community, so we will give

a brief description of the underlying physical principles.

Let us consider again the simplest case of a 2D free electron gas at constant

chemical potential, cf. section 3.1.2. In figure 11, we have sketched the density of

(orbital) states near the Fermi edge. The spacing of the energy levels is EhheB=m,i.e. a higher electron mass leads to a denser spacing.

The dHvA oscillations will only occur if the Fermi edge is sharp on the scale

of the energy spacing, i.e. for temperatures lower thanhheB=kBm. In fact, thetemperature dependence of the dHvA signal acquires the characteristic LifshitzKosevich damping factor

RT Xsinh X

where X 2p2kBTm

?

ehhB 15

which rapidly goes to zero for elevated temperatures.

For real metals, the thermal damping of the dHvA amplitude in this way encodes

information about the quasi-particle thermodynamic cyclotron masses m?. We will

discuss the effect of interactions and the different measures of quasi-particle mass in

section 3.4.Impurity broadening of the orbital energy levels also induces a washing out

of the Fermi edge traversal. Crudely speaking, dHvA oscillations will only be

observable if the level widthhh=2is smaller than the level spacing (see the lower panel

of figure 11; is the relaxation time). This leads to an additional damping factor



22/87

RDexpBD=B where BD hh2e

CFlfree

16

i.e. with a characteristic field BD that can be recast in terms of only the mean free

path lfree, taken to be constant, and the geometry of the Fermi surface: CF denotes

the circumference of the orbit in k-space. In real space terms, the damping factor

can be re-expressed as RD expCr=2lfree, where Cr is the circumference of thecyclotron orbit in real space.

3.2. Angle-dependent magnetoresistance oscillations3.2.1. Introduction

In quasi-2D metals, the magnetoresistivity for currents along the c-axis shows

a strong dependence on the direction of the applied magnetic field. This angle-

dependent magnetoresistance oscillation (AMRO) effect was first observed in the

organic metals b-(BEDT-TTF)2IBr2 [33] and -(BEDT-TTF)2I3[44] and subsequently

explained [34, 45] as being due to magic angles in the Fermi surface geometry. In this

way, the exact form of the AMROs constitutes a very subtle probe of the shape of

the Fermi surface, i.e. both the in-plane Fermi contour and the c-axis dispersion.

In the most basic case, a simple corrugation of an otherwise cylindrical Fermi

sheet exhibits maximum magnetoresistance at exactly those (Yamaji) angles atwhich all Fermi surface cross-sectional areas become equal (cf. section 5.4.1). At

these angles, the c-axis Fermi velocity averages to zero over each cyclotron orbit

(see figure 12), and we will see that this indeed suppresses the c-axis conductivity in

a strong magnetic field.

Energy

low mass,wide bands

high mass,

low temp.

high temp.narrow bands

impurity

broadening

Figure 11. Temperature and impurity broadening of the dHvA oscillationsfor details,refer to the text.



23/87

3.2.2. Semiclassical DC magnetoconductivity

In a more rigorous approach, we can understand AMROs within the relaxation

time approximation for the DC magnetoconductivity, as discussed in standard solid

state physics textbooks such as that by Ashcroft and Mermin [46]. This approach

views the electron movement from a semiclassical point of view, disregarding the

orbital Landau tube state and taking instead the wavenumber k as an almost good

quantum number which slowly revolves around the cyclotron orbit under the

influence of a magnetic field. During this motion, the electrons are scattered fromtime to time, and this scattering process is taken to be random ([46], p. 244). AMROs

are therefore subject to impurity damping, but to much less thermal damping than

dHvA oscillations.

As shown in [46], p. 248, the magnetoconductivity tensor is then given by

ij e4p3B

IFS

d2k

Z kcyclotron

orbit

dk0uuFikuuFjk0 Pk0, k

sin ffB, uuFk0 17

where uuFikis the ith component of the unit vector uuFkalong the direction of theFermi velocity at k. Here, the probability Pk0, k that a quasi-particle survives themovement along the cyclotron orbit from k0 to k without collisions is given byPk, k 1 and

dP

dk0Pk, k0 hh

eBlfreesin ffB, uuFk0 : 18

Again, the constant mean free path approximation is assumed.

3.2.3. Quasi-periodicity

First, we will generally deduce how equation (17) leads to an AMRO signal that

is quasi-periodic in tan 0; see e.g. the right panel of figure 12.Consider an arbitrary quasi-2D Fermi surface of a clean crystal with the angle 0

of the applied magnetic field close to 908, i.e. with high values of tan 0. Here, clean

is supposed to mean !c1; in equation (17), we can therefore effectively setPk0, k 1, to obtain

k

uF uF z

Yamaji Angle

()z

k

B

Figure 12. For magnetic fields at the Yamaji angle, the c-axis Fermi velocity averagesto zero over each cyclotron orbit, leading to a suppression of the c-axis conductivity.The labelling follows the notation of equation (17).



24/87

zz/I

FS

d2k

Zcyclotron

orbit

dk0uuFz k uuFz k0 Z hBZ

0

dkz0

Zcyclotron

orbit

dk uuFz k" #2

: 19

Here, the cyclotron orbit in question is the one cutting the cylinder axis atkz0 . As0is large, the cyclotron orbits extend vertically across several Brillouin zones, so thedirection uuFz of the c-axis Fermi velocity oscillates rapidly over such an orbit. The

integral in square brackets in equation (19) can therefore be approximated by its

stationary value, to which every warping component at periodicity gives

a contributionZ . . .

uuFzktop p=4 hBZ=2p uuFzkbottom p=4 hBZ=2p 20

times some curvature factor which is irrelevant herethe subscripts refer to the

highest and lowest value ofkz along the cyclotron orbit, respectively. Now ktop andkbottom are separated by ktan 0, where kk0 is the in-plane Fermiwavevector at the azimuthal angle 0 of the applied field. It immediately follows

that if we increase tan 0 by hBZ=k we will obtain the same stationary values inequation (20), the same integral in square brackets in equation (19), and hence a very

similarc-axis conductivity.

So AMROs are quasi-periodic in tan 0, and the periodicity interval is hBZ=k(provided that there are warping contributions at 1). AMRO measurements cantherefore directly infer the in-plane Fermi contour k0 if the material (a) is ofsufficient purity, and (b) possesses only one quasi-2D electronic band, since it is often

difficult to disentangle the contributions from several bands. Both conditions areoften fulfilled in quasi-2D organic molecular metals, and for them AMROs have

become a standard caliper probe of in-plane Fermi surface topography [47].

3.2.4. Analytical results for a circular Fermi contour

In general, the conductivity tensor integral in equation (17) can be solved

numericallywe will discuss how to do so effectively in section 4.3. The direct

comparison between experiment and the full computation of the AMROs expected

from a model Fermi surface is a very powerful tool, going beyond the mere mapping

of the in-plane Fermi contour via the periodicity of the AMRO maxima [48]. We will

utilize such computations in sections 5.3 and 5.7.2 to extract the Fermi surface

geometry from our high-precision experimental AMRO data on Sr2RuO4.

Still, it is quite instructive to evaluate the AMRO integral in equation (17)

analytically for the simplest cases. Some of these results are already known [45, 49];

here we will present a simple generalization.

We consider aclean(!c1) crystal withisotropicin-plane electronic structure,i.e. circular Fermi contour. Thec-axis conductivity in equation (17) then simplifies to

zz/ Z 2p

0 dz, 0 Z 2p

0 dZ 2p

0 d0uuFzuuFz0 21

uuFz /X

0>0

ksin zcos 22

z z, 0 Ftan 0cos 0 23



25/87

where only the cosinecosine terms in the cylindrical expansion in equation (2)

have been included for now; an extension to terms containing sines is trivial. The

integral can be solved analytically, step by step: first, the integration over z, 0necessitates 0 in the cross-terms. The sin z terms in equation (22) can beexpanded in terms of cosn 0 with the help of the expansion exp i cos x P

imJm expimx, where m runs from1 to1. Taking all cross-terms intoaccount, we arrive at

zz/X, 0 ,

0even

i02kk0cos 0cos

00JFtan 0J0 Ftan 0: 24

A few special cases are interesting, and we have sketched them out in figure 13:

For simple warping, we obtain

zz/J2

0 Ftan 0

25

giving AMRO maxima whenever J0Ftan 0 0, i.e. at the Yamaji angles.

Simple tetragonal cuprates are thought to have extremely anisotropic c-axistransport with warping nodes along the Brillouin zone diagonals [51] which we

can hope to characterize byk01 andk41 contributions of equal size.5 Neglecting

the effects of the non-circular in-plane Fermi contour in these materials,

we arrive at

zz/ J20 J24 cos2 40 2J0J4cos 40 J0 J4cos 402 26whereJ is shorthand for JFtan 0.

Figure 13. AMROs in the cases outlined in the text, in the limit of very pure samplesand for a circular in-plane Fermi contour. The diagrams are in tomographicrepresentation, with the azimuthal angle denoting and the radius denoting Ftan 0with a range from zero in the centre to 16 at the edge. Dark shading denotes smallresistance, bright denotes the AMRO peaks. the expressions for zz are given, whereJ is shorthand for JFtan 0. In the cuprate panels, it is evident how theperiodicity in tan 0 is only established at higher angles. The Tl2Ba2CuO6x panel usesthe warping parameters extracted in [50] and given in the text, including a smalldeviation k

40 from circularity, so that the expression in equation (27) is only a (very

good) approximation.

5This assumes coherent interlayer transport which might actually not be applicable in some

cuprates.



26/87

Most hole-doped body-centred tetragonal cuprates such as Tl2Ba2CuO6x arethought to have a hole cylinder near the X line as their Fermi surface, very much

like thecylinder in Sr2RuO4, but somewhat larger. The warping on that sheet

is expected to (a) predominantly involve nearest neighbour planes (1),between horizontally offset copper sites, (b) conform to the body-centredtetragonal symmetry, much like the sheet in Sr2RuO4, and (c) have warping

nodes for0 458, much like the case for the simple cuprates dicussed above. Ifthe three lowest symmetry-allowed nearest-layer terms are retained, the generic

warping is then proportional tok21sin 2 k61sin 6 k101sin 10 cos z,where we need k21 k61k101 0 to satisfy condition (c).The result in equation (24) can easily be extended to warping terms containing

sines; the result in this case is

zz

/ k21J2sin 20

k61J6sin 60

k101J10sin 100

2:

27

This expression differs from previously published predictions for AMRO

in Tl2Ba2CuO6x [52], showing that the warping symmetry and the speciallocation of the Fermi surface in the Brillouin zone corner have not been

treated correctly in that publication. Remarkably, Hussey and coworkers have

very recently managed [50] to observe such AMROs in Tl2Ba2CuO6x in>40 T magnetic fields, in what is probably the first transport evidence that

(overdoped) cuprates possess a coherent three-dimensional Fermi surface. The

warping parameters they extracted arek214:0,k61 2:7, andk101 1:1 aswell as k40

19 (each in units of 107m1).

3.3. Cyclotron resonance

In a similar fashion to AMRO, the microwave conductivity of quasi-2D metals

shows characteristic resonances in an applied magnetic field, indicative of both the

Fermi surface geometry and the quasi-particle mass renormalization. This subject

has been dealt with in great detail in the available literature, both for in-plane [53]

andc-axis [35] microwave conductivities, and here we will only present a summary of

the main ideas relevant to Sr2RuO4.

In the simplest case of an isotropic in-plane electronic structure, a magnetic field

applied along the c-axis induces cyclotron motion of the charge carriers around the(circular) Fermi contour. The microwave conductivity therefore shows a resonance at

the cyclotron frequency!c eB=mc, wheremcis the cyclotron resonance mass of thecharge carriers (which can, in general, deviate from the quasi-particle thermodynamic

cyclotron mass m?, see the discussion in section 3.4). The resonance sharpens as the

mean free path and the quasi-particle lifetime increase.

For non-isotropic in-plane electronic structures in general, and for non-circular

Fermi contours in particular, resonances in the in-plane microwave conductivity also

occur at odd multiples of !c [53]. Information about the in-plane Fermi surface

geometry is encoded in the relative strength of these resonances.

In the presence of out-of-plane corrugation of the Fermi cylinders ( >0), thec-axis microwave conductivity becomes finite and also shows resonances at integer

multiples of!c. For these resonances to occur, the warping needs to be non-trivial in

the sense that some k with >0 is non-zero, or the field needs to be tilted away

from thec-axis, so that the c-axis component of the Fermi velocity is modulated. In



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this way, the relative strength and the angular dependence of these resonances can be

related to the details of the warping of the Fermi surface.

Generally, all these predictions originate from an evaluation of the semiclassical

frequency-dependent conductivity ij! which, in the relaxation-time approxima-tion, can be expressed in almost identical fashion as in equation (17)except thatthere is an additional phase factor expi!tk0, k in the integral, where tk0, kdenotes the time it takes for a quasi-particle to traverse on the cyclotron orbit from

k0 to k.We will use these guiding principles for the quantitative analysis of microwave

resonance data on Sr2RuO4 in appendix A.

3.4. Quasi-particle masses

When dealing with a real metal, the electron mass can deviate from its bare value

m 9:109 1031

kg. The value of the mass is related to the bandwidth: simplesystems like the alkaline metals have electron masses close to m, while tightly bound

d electrons, e.g. in transition metals oxides, tend to have narrower bandwidths and

higher masses. Some intermetallic compounds on the basis of Ce or U, so-called

heavy fermion systems, can have electron masses exceeding 100m due to the

interplay between the s/p and f electrons [54].

The masses relate to one specific orbit in k-space and can change when a different

orbit, or a different direction of the applied magnetic field, is chosen. For a quasi-2D

system, the situation is simplified in that each Fermi surface sheet possesses one

well-defined mass for on-axis fields, and off-axis the masses go as 1 = cos 0.

In addition, the exact concept of the electron mass also becomes blurred in a realmetal, and depending on the context there are at least four different relevant electron

masses to be dealt with, affected in distinct ways by the electronelectron (ee) and

electronphonon (ep) interactions and relating to different material properties.

The band mass mb is usually taken to be the unrenormalized mass asdetermined by band structure calculations.6

The thermodynamic cyclotron mass m? is the mass measured in dHvAexperiments, and relates to the actual slope of the renormalized Ekdispersion, see figure 11. This also determines the specific heat, which in a

quasi-2D system must then match the Fermi surface mass enhancement via

C

T pk

2BNAa

2

3hh2

Xm?i1:48 mJ mol1K2

X m?im

: 28

It is affected by ep renormalization (cf. [46], p. 520) and by the (spin-

symmetric part of the) ee interactions (see below).

The cyclotron resonance mass mc is a measure of the frequency !ceB=mc atwhich thek-states precess around the orbit in a magnetic field. It is affected by

the ep interaction, but the ee interaction has less of an effect.7

6Such calculations are usually quite successful in predicting the overall bandwidth, but they fail to

describe the subtle renormalization effects at the Fermi level.7In a Galilean invariant system, this fact is known as Kohns theorem; see [55] and references

therein for an in-depth discussion.



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We also introduce here the susceptibility mass m?susc as a measure of theenhancement of the spin susceptibility, i.e. by how much in k-space the spin-up

and spin-down bands are driven apart in a magnetic field. The overall (SI

volume) spin susceptibility is

s 202B

phh2c

Xm?susc, i 4:45 106

X m?susc, im

29

assuming ag-factor close to 2.m?suscis affected by the spin-symmetric as well as

the spin-antisymmetric (Stoner) part of the ee interactions;8 it is a famous

result (cf. for example [46], p. 663), however, that ep interactions have no

effect on m?susc since they affect each subband separately at the Fermi level,

conceptually after the Zeeman splitting has already happened.

The influence of the various renormalization contributions on the different quasi-

particle masses is summarized in the upper part of table 11 in appendix B. For an

isotropic, Galilean-invariant (3D jellium) electron system, these relations can be

made more precise [55, 56]: quantifying the electronelectron interaction through the

Landau f-function and its expansion Fs=an into spin symmetries and spherical

harmonics, we have the famous relations m?=m 1 Fs1=31 , m?susc1 Fs1=3=1 Fa0 , and mc=m 1 . Here, is the ep renormalization termwhich usually takes on a value between 0 and about 1 ([46], p. 520).

4. Experimental techniques and analysis procedures

In section 3, we outlined the basic conceptual ideas behind the dHvA measure-

ment, and discussed the information that dHvA frequency and amplitude reveal

about the electronic structure of quasi-2D metals. In this section, we will briefly

discuss how to access this information experimentallythe motivation being that

(1) the classic textbook in the field [31] is out of print, (2) the 2f field modulation

technique, which is the most powerful dHvA measurement method, introduces

artificial structure into the dHvA amplitude (behaviour that is well-understood

but needs to be considered prior to further analysis), and (3) the amplitude analysis

and its numerical matching to model Fermi surfaces is a new concept.

9

4.1. dHvA techniques

4.1.1. Field modulation and cantilever deflection

The two most commonly used dHvA detection techniques are field modulation

and cantilever deflection. Field modulation is the traditional workhorse of dHvA

studies [31]: a modulation field of amplitudeb a few mT and frequencyf a few Hzis applied to the sample on top of the background field B0, and the voltage induced

at a pick-up coil around the sample is recorded (figure 14).

8In the nomenclature of Fermi liquids this refers to Fs1 and Fa0 , respectively, cf. later in this section.

9We will not discuss the technically challenging but conceptually straight-forward measurement of

AMROs, nor will we describe microwave absorption, ARPES, or any of the other relevant experiments

in any detail, since these are not the main focus of either the article or the authors expertise.



29/87

It turns out that the pick-up voltage is not purely sinusoidal in time but has

additional frequency components at the harmonics f, with an amplitude [31]

proportional to

RFM J2pFb=B20: 30

That is, if the modulation amplitude b becomes of the order of the repeat distance

B B20=Fof the magnetization profile, we can record the voltage at 2f to get asizeable and essentially background-free quantum-oscillatory signal. Of course there

are a few non-trivial issues involved in maximising the signal-to-noise ratio [57], butthese technicalities are beyond the scope of this article.

In contrast, the cantilever deflection technique works better in the noisy

environment of resistive high-field magnets and senses the quantum-oscillatory

torqueon the sample. For a quasi-2D system, the magnitude of the torque (density)

isMB sin 0, i.e. the method loses its sensitivity at low fields or near a symmetryaxis of the crystalbut otherwise it is quite useful, provided one takes adequate

precautions against the torque interaction effect [58, 59].

We have performed several thorough dHvA rotation studies, with the field

rotating in both the

001

! 100

and

001

! 110

planes, on different high-quality

crystals of Sr2RuO4 with Tc ranging between 1.3 K and 1.45 K. The experimentswere carried out both on superconducting magnet systems, in magnetic fields up to

18 T, and a resistive magnet at the NHMFL Tallahassee, in magnetic fields up to

33 Tthe former setup provides much superior noise levels, at the expense of raw

magnetic field. In both cases, we achieved base temperatures of 50 mK or lower with

B

M(t)M(B)

B

B(t)

t

t

Figure 14. Sweeping a sinusoidal modulation field Bt B0 b cosft over the quantum-oscillatory magnetization profile leads to an intricate pick-up signal with contributionsat f.



30/87

a dilution refrigerator. In all situations, the applied magnetic field completely

quenches the superconductivity, thanks to the low value of Hc2 (its maximum is

1.5 T for fields applied parallel to theab-plane, see [60]).

4.1.2. Sample sweep and beat pattern

A typical trace of the quantum oscillatory magnetization of Sr2RuO4 versus

applied field can be seen in figure 15, taken from [61]. The sample is a high-quality

Sr2RuO4crystal of dimensions 2:7 1:3 0:26mm3, with a superconducting criticaltemperature Tc1:435 K, prepared in a floating zone image furnace. The crystalwas mounted flat on the top plate of a phosphor bronze capacitive cantilever

torquemeter, and the torque on the sample was detected using a standard passive

capacitance bridge, driven at several kHz and connected to the torquemeter with

miniature coaxial cable. The magnetic field was generated by a resistive 33 T high-

field magnet at the NHMFL Tallahassee, and the sample was cooled to 30 mK on adilution refrigerator.

Figure 15 also displays the envelopes, or field-dependent amplitudes, of the

oscillations corresponding to the three fundamental frequencies. These amplitudes

5 10 15 20 25 30

cAxis Field (Tesla)

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

Magnetization(a.u.)

sheetsheet

sheet

0 5 10 15 20

DHvA Frequency (kT)

1

10

AmplitudeSpectrum(a.

u.)

23

Figure 15. Quantum-oscillatory magnetization of Sr2RuO4 in a long field sweep from 33 Tdown to 2 T (thin grey line), at 30 mK with the field 4:58 off the c-axis. The datahave been numerically compensated for the torque interaction effect. The dHvAfrequency spectrum, taken over the field range 25 T! 33 T, is given in the inset. Thethick black lines show the envelopes of the oscillations corresponding to the threefundamental frequencies. (Reprinted from [61] with permission from Elsevier 2001.)



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are extracted from the data by filtering in the Fourier domain, according to the

scheme presented in section 4.2.

One clearly observes beats in all three envelopes of the oscillations correspond-

ing to the three respective Fermi surface sheets. These beats originate from the

interference pattern induced by the corrugation of the Fermi cylinder into themagnetization in equation (8).

4.2. Envelope extraction

When analysing the dHvA results in this paper, we are often interested in the

field-dependentamplitude of the oscillations, rather than the dHvA frequency itself.

To be specific, if the experimental data are given by oscillations at well-separated

frequenciesFn

~MMB Xn

AnB sin 2pFnB

31we want to extract the envelopes AnBof the individual oscillations. The envelopesare assumed to vary slowly on the scale of B2=F, where F is the frequency

separationthis means that the peaks in the dHvA spectrum do not overlap.

The dHvA (Fourier) spectrum ~MMF, taken against inverse field, is given by

~MMF 12i

Xn

AnF Fn AnF Fn : 32

As the envelopesAnB are slowly varying, their Fourier transformAnF is narrowlypeaked around zero. Correspondingly, equation (32) describes a spectrum with fairly

well-defined peaks at the dHvA frequencies Fn (which is not a surprising result). We

can now easily extract the envelopes An byfilteringthe dHvA spectrum atFn, that is

Fn only, which leaves only the contribution

~MMfilteredF 12i

AnF Fn 33

and therefore, reverting back to (inverse) field space,

~MMfilteredB 12i

AnB exp 2piFnB

34which, finally, gives the envelope as

jAnBj 2j ~MMfilteredBj: 35We can therefore extract the envelope by filtering the dHvA signal around Fn in the

dHvA frequency domain, converting back to (inverse) field space, and taking the

modulus.

The correct choice for the filter width F is quite crucial: it needs to be wide

enough to accommodate the features in AnF; in particular, for warped Fermisurfaces, it has to be larger than the beat frequency. Also, the data region near the

ends of the field sweep, where the extracted envelope will be affected by artificial

windowing effects, is on the order ofB2=F, so that a wide filter provides meaningful

results over a wider field range. On the other hand, F obviously needs to be smaller



32/87

than the frequency separation F, and generally narrower filters give better signal-

to-noise in the envelope function. In practice, for the frequency spectrum of

Sr2RuO4, a filter half-width of around 0.4 kT turns out to be appropriate.

4.3. Numerical approach to AMRO, cyclotron resonance, anddHvA in quasi-2D systems

In general, the theoretical AMROs, microwave conductivities, and dHvA

amplitude pattern have to be calculated numerically from a given Fermi surface

parameterization. We will discuss how to do so efficiently, since our aim is to

successively refine the parameterization through comparison with the experimental

data.

The dHvA amplitude can be integrated from equations (8) and (9) in a

straightforward manner. In the spirit of section 4.2, the envelope can be extracted as

AB X"#

Z 2p0

dzexp ihha"#zeB

: 36For extremely small warping, the danger of round-off errors in the cross-sectional

area acan be circumvented by a differential technique, using the deviation a from

the non-corrugated areas, but this is not necessary for the moderate warping found

in Sr2RuO4. The integral usually converges fast unless the warping massively exceeds

the Landau tube spacing (i.e. at small fields). In that case, however, the conventional

extremal orbit formulation becomes valid again, and the full integration is not really

needed anyway.

AMRO requires more computational effort. First, one needs to evaluate theprobability Pk, k0 that an electron completes an orbit from k0 to k withoutcollisions. In the constant mean free path approximation, equation (18), this is

given by

Pk, k0 exp hhKarck, k0

eB?lfree

37

where Karck, k0 is the in-plane arc length of the Fermi contour between k and k0.Note that this expression doesnot depend on the magnitude of the Fermi velocity or

the quasi-particle renormalization; just like the magnetoresistance itself, it is purely afunction of the Fermi surface geometry and the mean free path. This is because

heavier electrons travel more slowly and therefore scatter less frequently and are less

influenced by the Lorentz force.

The integral over the cyclotron orbit in equation (17) also includes those paths

between k 0 andk where the electron winds more than once around the orbit. Such asemi-infinite integral is impractical to calculate, and since the probabilities Pk, k0 ineach case follow a geometrical series, we can replace equation (17) with

ij

e

4p3B? IFS d

2k Icyclotronorbit dk0uuFi

k

uuFj

k0

Pk0, k1

P2p

38

where P2p is the probability that an electron traverses round the cyclotron orbitonce without scattering.

The unit vectors uuF can be calculated geometrically from equation (2), and we

can finally perform the integration rather quickly. For the integration, it is useful and



33/87

memory-saving to evaluate the integrand in equation (38) at points on the cyclotron

orbits which are equally spaced in terms of the in-plane arc length between them,

since the evaluation of equation (37) is then greatly simplified. In practice (i.e.

for Sr2RuO4), one rarely needs to go beyond 512 points to calculate zz to 0.1%

precision for 0 up to808.The computation slows down at high angles 0 when uuFz oscillates rapidly along

the cyclotron orbit. However, for 0908, the situation simplifies again, as thecyclotron orbits are now straight lines running down the Fermi cylinder. It is then

possible to perform the integral over k0 analytically, and equation (17) reduces to asimple integral over the in-plane Fermi contour. However, it is not particularly

instructive to present the details here, especially since this approach neglects the

additional effects of the small closed cyclotron orbits which are theoretically [62] and

experimentally [11] known to dominate thec-axis magnetoresistance in Sr2RuO4for

high in-plane fields.

The microwave conductivity, finally, can be calculated in complete analogyto AMROexcept that now we need to deal with the additional phase factor

expi!tk0, k. The orbit traversal time tk0, k does depend on the quasi-particlerenormalization; this is why microwave conductivity resonances can yield the

cyclotron resonance mass mc. In a simple model, one can assume tk0, k /Karck, k0, but this does not necessarily have to hold, especially for the orbit inSr2RuO4 where the cyclotron motion can be expected to slow down near the van

Hove singularity. In any case, thankfully, the geometric series trick for multiple

winding orbits can also be applied to the phase factor expi!tk0, k.

5. Experimental results and analysis

The quantitative extraction of the Fermi surface parameters of Sr2RuO4rests on

extensive data sets obtained in dHvA and magnetoresistance studies, combined with

the measurement of bulk thermodynamic parameters, and the use of photoemission

and microwave resonance data. We will start with a review of the experimental

confirmation of the basic Fermi surface topology and then proceed to refine our

view until we arrive at a full geometric description. We then consider the mass

enhancement and apply a few consistency checks. Finally, we will present and review

a few recent studies of how the electronic structure changes in response to

hydrostatic pressure.

5.1. Basic properties

Measurements of the bulk heat capacity C=T, Pauli spin susceptibility s,

resistivity, and Hall number RH in Sr2RuO4 provided early indications that

Sr2RuO4is an ordinary metallic Fermi liquid, albeit a very anisotropic one and with

strongly renormalized electron masses. The experimental values are listed in table 3.

Here, the measured bulk susceptibility [63] is not exactly equal to the Pauli spin

susceptibilitys but contains additional spin-orbit, van Vleck, and Landau diamag-

netic contributions (see, for example, [64]). While it is generally believed that these

susceptibility contaminations are relatively small in Sr2RuO4, due to the large mass

enhancement, NMR Knight shift measurements [65] give a spin susceptibility value

somewhat lower than the b

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Berg Em Anna Dv in Phys 2003